Solutions and comparison of Maximum Likelihood and Full-Least-Squares estimations for circle fitting

The fitting of a number of noisy data points with a circle has found numerous applications in image processing and pattern recognition. This paper examines two methods to estimate the circle parameters: the Maximum Likelihood (ML) method and the Full-Least-Squares (FLS) method. The ML method is based on the noisy model from the data while the FLS method minimizes the geometric distance square. We first provide the iterative solutions of them using Taylor-series linearization approach. We then show analytically that FLS does not yield the ML solution. This is in contrast to previous study that the FLS method gives the same solution as ML. FLS method approximates the ML estimation only if the noise power is much less than the circle radius square. Simulations are included to support the theoretical development.